Bayes´ optimum filters derived using Wiener canonical forms

Canonical forms and circuit structures are derived for Bayes\´ optimum decision rules utilizing Wiener-like functionals. The formalism is valid for loss functions whose derivatives with respect to the estimates exist, ( over the space of unknowns ), The structures derived consist of orthonormal filter sets (usually Laguerre filters ) followed by combinations of -law-devices (Hermite polynomial generators of the Laguerre coefficients), amplifiers of specified gains, and then summing and division circuits. The amplifier gains can be partially preadjusted via a sample of the observable and unknown or by a pre-computation; the remainder of the gain adjustment is, in the general case, obtained via feedback through a function generator. If the gains are adjusted via the first method, ergodicity is required and self-adaptive features are implied. As a first step in the exposition, analogous canonical forms and structures, where the past of the observable is characterized by its time-samp]es rather than by its Iaguerre coefficients, are obtained. The derivation does not stop here because of the limitations of the description, analysis, and equipment-realization of a stochastic process in terms of a finite number of its time samples. The advantage of the structural forms derived over previous ones is that they are formally valid for all values of signal-to-noise ratio and are always physically realizable and time-invariant whereas this has not usually been true in the past. Several examples of structures are given. Existence and convergence problems are discussed in the appendices. Perhaps more important than the explicit results obtained here are the implications involved in the procedure. The fact that expansions of the probability measures for a significant class of stochastic processes has been obtained in terms of canonical expansions of the Wiener process is felt to be a major accomplishment.